Moduli spaces of bundles and Hilbert schemes of scrolls over $\nu$-gonal curves

TL;DR

This paper improves the classification of certain vector bundles on general -gonal curves and explores the properties of related Hilbert schemes of surface scrolls, revealing their stability and special features.

Contribution

It advances the understanding of Brill-Noether loci for rank 2 bundles on -gonal curves and analyzes the stability of associated surface scrolls in projective space.

Findings

Enhanced classification of irreducible components of Brill-Noether loci.

Identification of stable and special properties of Hilbert schemes of scrolls.

New insights into the geometry of vector bundles and surface scrolls over -gonal curves.

Abstract

The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $\nu$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $\nu$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.